hmsloop_hotspur | Navigatio Britannica: or, A Complete System of Navigation (Reply)

On the poop Crystal with his sextant was patiently trying to instruct the midshipmen in the elements of navigation — the young devils were fidgety and restless as Crystal droned on. Hornblower was sorry for them. He had delighted in mathematics since his boyhood; logarithms had been playthings to him at little Longley's age, and a problem in spherical trigonometry was to him but a source of pleasure, analogous, he realised, to the pleasure some of those lads found in the music which was so incomprehensible to him.
— A Ship of the Line

Back in July I started reading John Barrow's Navigatio Britannica: or, A Complete System of Navigation, a popular eighteenth-century text on navigation, largely because I wanted to find out about this "spherical trigonometry" that gets mentioned every now and again in the Hornblower novels. (I wanted to know just what it is that Bush has nightmares about!) I had hoped to be farther along in it by now, but I'm about halfway through, and what with August almost over, I thought I'd post a round-up of random observations on the first five chapters for the Mathematics prompt.

I. Elements of Chronology

The first thing that blew my mind: seconds are called that because they are literally the second time you divide something (such as hours or degrees) into sixtieths. (The first time of course being minutes, from the Latin minuto primo, literally the FIRST time you divide a thing into minutes!) Much to my shock, there are further units after seconds called thirds (a sixtieth of a second), fourths (one three-hundred-sixtieth of a second), and so on.

Most of this chapter is spent on how to read a calendar, because instead of a nifty calendar being available free at the local drug store each year as a promotional item, pre-printed with days of the week and phases of the moon, calendars are what we would today call a perpetual calendar, published once and good forever. For instance, instead of being labelled with the days of the week (and thus only being good for that year), the days of the week are labelled A-B-C-D-E-F-G, and you have to calculate for yourself -- a nifty little formula is given -- if this is a "Sundays are on A" kind of year or "Sundays are on B" or what. Same with phases of the moon: they're marked on the calendar in super-generic ways, and formulas are given for figuring out where full moons and new moons (the latter called "change days") fall with respect to the generic markings this year. Once you know that, further formulas are given for calculating the time of the moon's "southing" -- the moon's highest point in the sky, its own "noon," so to speak -- given today's phase.

By the way, the second thing in this chapter that blew my mind: Leap Day is not February 29th, but inserted between February 23rd and 24th, such that Feb 24 was forty-eight hours long. According to wikipedia, this went back to Julius Caesar's reform of the calendar: the new year was March 1, they wanted to slip the leap day in as close as possible to that, but February was a brutal month of multi-day festivals -- but there was a gap in the festival schedule at Feb 23/24, and so that's where the leap day went! Apparently that's still where it fell circa 1750, when this book was published; when it switched over to Feb 29, I don't know.

II. Of the Flux and Reflux, or the Ebbing and the Flowing of the Sea

Construct your own tide tables! Because again, you can't just go down to the local bait shop and pick one up for the month.

Remember the formulas for calculating the phases of the moon and the time of the moon's southing? For any locality, high tide occurs at about the same time relative to the moon's time-of-zenith, all month long. (E.g., if high tide precedes the moon's "southing" by three hours during the full moon, it'll precede it by three hours at the quarter moons and "change days," too.) However, Barrow doesn't do any of this in hours, he does it by compass points: if high tide is when the moon is SE on one day (during its path through the sky from east to south to west again), then high tide will always happen when the moon is SE. A list is given noting the compass direction for high tide at various important localities: at Brest, high tides are when the moon is at NE by E and SW by W; at Portsmouth, they are when the moon is at N and S. There's also a fold-out conversion chart so you don't have to do the math yourself: just look up today's moon phase and the relevant compass points for the place that interests you, and it gives that day's high tides.

I spent a little while playing around with this, predicting tides at various places and comparing them to online tide-prediction websites: at full and new moons it tended to be very accurate, at quarter moons it might be forty-five minutes off. Pretty good for a simple universal tide table, I thought.

III. Geometry

Euclid's greatest hits! Taught in the old-fashioned style, with compass-and-straight-edge constructions! Honestly, if you remember 10th-grade geometry, you probably already know this chapter.

One curiosity that struck me: while defining various shapes, they eschewed the word rectangle and referred to them as right-angled parallelograms, which I had a bunch of ???? feelings about: Do rectangles not matter in 18th-c. geometry? But it turned out that they were reserving the word for multiplication! You know how a square of a number is that number times itself? Well, a rectangle of two numbers is those numbers times each other! Exactly parallel in usage to square!

And in Trivia Questions You'll Never See in a Pub Quiz, have two words that appear to have died with the 18th c.:

amblygonum – an obtuse triangle
oxygonium – an acute triangle

Btw, I also learned while researching this chapter that trapezium has different definitions in British and American English – in Britain, a trapezium has one pair of parallel sides, while in North America, a trapezium has zero pairs of parallel sides. In this book, circa 1750, trapezia includes both classes of figures, which makes it easier to see how that split happened!

IV. Trigonometry

Dude. Dude. I can understand why this chapter gave Bush night sweats while he was swotting for his Lieutenant's exam. DUDE.

Honestly, the first half of this chapter wouldn't be nearly so bad if Barrow didn't make a side-trip into calculus and binomial expansions in order to prove the circumference of a circle so that he could then tell you how to calculate a full trigonometric table of sines and cosines, minute-by-minute, from scratch. (I'm serious. Estimate the sine of 0-deg 1', then use the pythagorean theorem to get the cosine, then calculate the sine of 0-deg 2', then its cosine, then the sine of 0-deg 3'...) But he does. He very, very much does. And this is after he gives a reassuring little speech about how he's going to make trig as straightforward as possible. I don't even know, man. But now that I know how one calculates a table of sines and cosines from scratch, all I can hope is that the poor sods who had to do it got paid very very well.

Also, there were typos in some of these proofs and corollaries. Typos that weren't in the errata sheet. I lost a full day to figuring out the typos, and I already knew planar trigonometry.

ANYWAY. I sympathize with Bush, is all I can say.

But once you get past all the theoretical setup -- which is there solely to teach you how to derive trigonometric tables on your own! -- the planar trig section isn't too bad. There are lots of example trig problems, with each calculation performed three ways:

Draw the triangle to scale, lay your chart dividers/compass on it, and just fucking measure what that last side/angle is. Fussy, but straightforward! Almost like crafts-hour!
Use your trig formulas and trig tables, multiplying or dividing as necessary, just like you learned in high school (assuming that you're old enough that you did it with tables instead of on the calculator). Honestly, this method looks like its own special hell, given that it all has to be done longhand. (Although it's a bit better if you use logarithms, but logarithms are in the next chapter.)
Magic, aka Gunter's Scale.

I had to look up what this Gunter's scale was, and I am in love: it's basically a slide-rule, but with scales specifically designed for ship's officers to work navigation problems on. (I say basically a slide-rule: a single piece, so no slidey action -- instead of sliding the two scales you're interested in against each other to see how they line up, you use your chart dividers to measure off your interval from one scale onto the other. Same method, different action.) From the way the text describes using it, I have to assume that the trigonometric scales are actually log-trig scales, but it makes these planar trig problems as easy as measuring one interval with your chart dividers on one scale, laying it on a second scale, and reading off the answer. Magic!

(I WANT A GUNTER'S SCALE SO BAD.)

Anyway, these things were apparently ubiquitous among ship's officers, and I can see why. With one of these, planar trigonometry is a snap!

The second half of the chapter is spherical trigonometry, and I admit that I cheated: I spent a fair amount of time with Todhunter's 1880's college textbook, Spherical Geometry. Because unlike with Barrow, Todhunter still had its diagrams (the scan of Barrow didn't fold out any of the pages that contained the figures, so they were mostly invisible). Also Todhunter gives proofs, and I often don't understand a thing until I've seen the proof.

The main difficulty with spherical trigonometry, now that I've messed around with it, is that you have to visualize things in three dimensions, which is more work than visualizing things in two. Worse, it doesn't all take place on the surface of a sphere: there are TWO angles opposite any given side of a triangle -- one on the surface of the sphere, analogous to the opposite angle in planar triangles, but one also at the center of the sphere, which describes the length of that side. (I.e., a side going from 30-deg N latitude to 45-deg N latitude is fifteen degrees long, as measured by an angle from the center of the sphere.) With spherical trigonometry, it's also much harder to sketch out anything that's confusing you than it is with planar geometry: at one point I was sketching in pencil on the leather surface of a baseball, trying to understand what a textbook meant by a thing. But of course even then you can't sketch it all out, because some of the angles are happening inside, under the leather cover, at the center of the baseball.

The formulas for spherical trigonometry are also more complicated than they are with planar geometry. For example, the interior angles of a triangle don't sum to 180, they sum to somewhere between 180 and 540, it depends. And it goes on like that. There's a certain beauty to it all -- I sincerely enjoyed teaching myself spherical geometry -- but spherical triangles have a lot more going on than planar triangles. I can absolutely see Bush being able to do it well enough to captain the Nonsuch, especially with reference to cheat-sheets and custom tools like a Gunter's scale, but I can also see how he was never comfortable with it.

Anyway, if you'd like to look at some pretty pictures about how spherical geometry relates to navigation, but don't want to have to read too much text, I recommend this website here: Sail Away: Notes on Plane and Spherical Trigonometry.

Oh, and here's a bit of trivia I learned in this chapter, which has bugged me since I was a wee thing big enough to read a nautical chart for myself: a nautical mile is the distance of one minute at the surface of the Earth. (Well, it depends on what direction you go: the Earth isn't a perfect sphere. But roughly speaking...) The nautical mile isn't there because imperial units are needlessly complicated! It's designed to make navigation problems simpler! It's almost like doing physics in metric!

V. Of Logarithms

Once again, Barrow drags out his higher maths to let us derive logarithmic tables on our own, once again making everything needlessly complicated. (Dude, you published log tables in the book; we're not going to derive our own!) He also claims that Napierian logarithms are the simplest and sweetest things, and I don't know what Barrow was smoking there, because the whole point of Briggsian logarithms -- also known as base-ten or common logs -- is that they're simpler to work with than Napierian logs. (Seriously. Napier invented logs, and then Briggs, who was interested in navigation, came along a decade or two later and made them straightforwardly practical.)

Anyway, the whole point of logs is that they change multiplication and division problems into addition and subtraction problems: no need to do long multiplication or long division ever again! This is the principle by which slide rules and Gunter's scale work, and it is a HUGE time-saver for people who have to do all their calculations by hand: adding two six-digit numbers is MUCH simpler than multiplying them.

btw, Barrow doesn't cover this, but before logs were invented, one of the off-label uses of trig was to get out of having to do multiplication. For example, this formula can be used to cheat your way out of multiplication:
cos(alpha)cos(beta) = (cos(alpha + beta) + cos(alpha - beta)) / 2
Have two numbers you want to multiply? Pretend they're both cosines and look up their "angles" in the table. Add those angles and look up the cosine; subtract those angles and look up the cosine. Add those two cosines and divide by two: voila, it's what you would have gotten if you'd multiplied your original numbers, but without having to do any multiplication at all!

Because multiplication is that evil, that you'd rather do trig instead!

And voila! That's the mathy first half of Navigatio Britannica! If you're interested in more detail, I liveblogged it on tumblr, and of course I'm always happy to nerd out in comments!